Recurrence & Induction:
Demonstrate by induction on n the following property :
(question in attached file)
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3. Consider the following recursively defined sequence: 4 = aut = : (n + 2) (for neN) . Prove, by induction, that an for all neN.
t= 1 1 - cos (2n.) 11. Prove by induction that sin (2t - 1)2) = = for all n e N and all r ER where sin : +0. 2 sin You may use the following two identities but you do not have to prove them. (5 marks) 2 sin Asin B = cos(A - B) - cos(A+B) cos (2A) = 1 - 2 sin? A
(4 marks) 3. Let the sequence {an} be defined by aj = 1, Q2 = 3 and an = 2an-1 - An-2 for n > 3. Consider the following statement to be proved by strong induction. an = 2n - 1 for all n EN Verify the base case(s) and carefully state the inductive hypothesis. You do not need to complete the proof here but may want to think it through before answering parts (a) and (b). (a) Base Case(s): (b) Inductive Hypothesis:
